Smooth-curve function generator



June 16, 1959 J. J. STONE 7 2,890,832

SMOOTH-CURVE FUNCTION GENERATOR Filed sin. 2. 1954 v 4 Sh gts-Sheet 1 l6 -W lf {F'. T 2 I0 15 I81 ZZZ-g INVENTOR.

Joseph J. Stone ,&W I

Juno: 16, 1959 J. J. STONE SMOOTH-CURVE FUNCTION GENERATOR 4 Sheets-Sheet 2 Filed. Sept. 2. 1954 INVENTOR. Joseph J. Stone DBY by ,Wm, i

ATTORNEYS.

SMOOTH-CURVE FUNCTION GENERATOR Filed Sept. 2, 1954 4 Sheets-Sheet 3 .Fl g 1U I NVENT OR. Joseph J. Stone BY b 6 I My,Mw, Dam

ATTORNEY,6.

June 16,- 1959 STNE 2,890,832

SMOOTH-CURVE FUNCTION GENERATOR Filed Sept. 2, 1954 4 Sheets-Sheet 4 ME "I IN VEN TOR. Joseph J. Stone WI J D -M ATTORNEYS.

2,890,832 SMOOTH-CURVE FUNCTION GENERATOR Joseph J. Stone, Columbus, Ohio, assignor, by mesne assignments, to the United States of America as represented by the Secretary of the Army Application September 2, 1954, Serial No. 453,870 2 Claims. (Cl. 235-197) This invention relates to a system for electronically developing nonlinear mathematical functions. More particularly, the invention relates to a system of developing a smooth-curve, nonlinear function by the use of diodes or other conduction-limiting devices in combination with a source of alternating or constant frequency potential.

Function generation has been accomplished in the past by various circuits utilizing diodes. In a basic circuit of this type, an input signal is applied to a series arrangement of a diode, a battery or other source of direct current, and a resistance. The output of the series circuit is then connected to a negative feed-back amplifier. In an arrangement of this type, the output signal will not appear until the input signal is at least equal in potential to the voltage of the battery and opposite in polarity thereto in respect to the series arrangement. The characteristic curve of output voltage versus input voltage is substantially linear. The slope of this curve is proportional to the ratio of feed-back resistance of the negative amplifier and the aforementioned series resistance.

Many modifications may be made in this basic circuit in order to permit the output signal to approximate the desired function of the input signal. These modifications, in general, build up a series of short, straight-line segments to approximate the desired function. Thus, if a biased diode is connected between ground and some point along the series resistance, the output signal will vary linearly with respect to the input signal until the diode conducts, at which time no further variation will appear in the output signal with increases in the input signal. If a resistance is placed in series with the lastmentioned diode, the output signal will vary linearly with respect to the input signal not only before the point of conduction of the diode is reached but also when the diode is conducting. The slope of the function when the diode is conducting will be less, however, than that when the diode is not conducting. In another variation, the output signal may be made proportional to the time integral of the input voltage by using capacitive feedback in the negative amplifier, or the output may "i tent be made proportional to the time derivative of the input signal by replacing the series resistance with a capacitance.

More complex functions may be generated by paralleling the series circuit with one or more similar circuits having different characteristics. These dilference characteristics may be obtained by varying the magnitudes of the sources of potential or by varying the magnitudes of the series resistances. Many other modifications may be used in these circuits to build up the desired function. In each case the developed function is built up of a series of short-time segments approximating the desired function. While in many instances such approximation will not seriously impair the use of the circuit, there are certain difiiculties inherent in the circuits that prevent their use in many other applications.

Since the function developed by these generators is a series of straight-line segments, the first derivative with respect to time of the function will appear as a series of steps and the second derivative with respect to time will appear as a series of pulses. Thus, with these generators, the second time derivative of the input function would serve no useful purpose.

To remove this difficulty requires a function generator that represents the desired function by curved sections connected in such a manner as to produce a continuous first derivative. This may be accomplished to a degree by increasing the number of diodes and utilizing the curvature of the diode characteristics. An improvement in the second derivative may also be obtained by shunting the feed-back resistance with, sufficient capacity to force the function to lag in time and, as a result, follow a curved path. The first of these methods involves a large number of tubes and extreme sensitivity to variations in operating temperature, while the second method introduces an excessive phase error.

The system of this invention utilizes a circuit similar to those previously described with the addition of a high-frequency, constant frequency signal connected in series with the input voltage. The constant frequency signal causes a current to exist which, when averaged over a cycle of the constant frequency signal, differs from that which would exist by virtue of the input voltage alone. Capacitance is added across the feed-back circuit of the amplifier to filter the high-frequency component of the input current without seriously affecting the output signal. The output voltage of this function generator is proportional to the average current produced. By proper selection of the wave form of the constant frequency voltage, the average current may be made to follow a curved path rather than a linear straight-line path, which it would have followed in the absence of the constant frequency signal. Thus, the function generator of this invention provides an output voltage that is a nonlinear function of an input signal applied to the generator.

A primary object of this invention is to provide a smooth-curve function generator.

Another object is to provide a function generator having an output that is a nonlinear function of an input applied to the generator.

Still another object of this invention is to provide means for developing a smooth-curve function in a function generator by introducing a constant frequency signal to the generator.

A further object is to electronically develop a smoothcurve function usable in function generation systems, such as electronic multipliers and the like.

Other objects and advantages of this invention will become apparent from the following specification, the accompanying drawings, and the appended claims.

In the drawings:

Fig. .1 is a partial schematic diagram illustrating the basic circuit embodying the principles of this invention;

Fig. 2 is a graph of output voltage against the input voltage for Fig. 1, illustrating the principle of this invention;

i Fig. 3 is a typical characteristic curve of the circuit shown in Fig. 1;

Fig. 4 is a characteristic curve showing the dependence of the output voltage upon the wave form of the constant frequency voltage introduced in the circuit of Fig. 1;

Fig. 5 is a schematic diagram illustrating a modification of the circuit of Fig. 1;

Figs. .6-A, 6-B, and 6-0 illustrate, respectively, the function generated by the circuits of Fig. 5, the first derivative of this function and the second derivative of this function;

Fig. 7 is a schematic diagram illustrating one method of building a function utilizing the principles of this invention;

Fig. 8 is a characteristic curve illustrating the function produced by the circuit of Fig. 7;

Fig. 9 is a schematic diagram illustrating a parabola generator utilizing the principles of this invention;

Fig. 10 is a curve illustrating the characteristics of the circuit of Fig. 9; e

. Fig. 11 is a schematic diagram showing an alternative circuit for parabola generation, and

Fig. 12 is a schematic diagram illustrating the circuit of an electronic multiplier utilizing the principles of this invention.

Referring now to Fig. 1, the basic circuit embodying the principles of this invention consists of an input terminal 10, a source of constant frequency voltage 11, a battery or other source of potential 12, a diode 13, and a resistance 14, connected in series with a feed-back amplifier 15. The amplifier 15 contains a feed-back resistance 16 shunted by a capacitance 17. An ouput terminal 18 is provided at the output of the amplifier.

The principle of operation of the basic circuit can be best explained by reference to Fig. 2. If a curve is drawn of the output voltage (2 of the circuit as a function of input voltage (e to the circuit without introducing a constant frequency voltage, the function will appear as a substantially straight line 20 having a slope equal to the ratio of the feedback resistance 16 to the series resistance 14. The output voltage will cut oif, however, until the input voltage (e is at least equal to the voltage 21 of the biasing battery 12 at which time the diode will conduct. Thus, if a voltage 22 less than the biasing voltage 21 is applied to the input terminals, no signal will appear at the output terminals. Now, if a constant frequency voltage 23 having a peak greater than the difference between the input voltage 22 and the bias voltage 21, is introduced in the circuit, an ouput signal 24 will appear. The high-frequency component of the output signal 24 is filtered by action of capacitor 17 to give a resultant average DC. output voltage 25. Thus, the output voltage will not have a sharp cutoff at the point where the input voltage is equal to the bias voltage. When the constant frequency voltage is held at a constant amplitude, and has a frequency substantially greater than the highest frequency applied to the input terminals, the characteristic curve of output voltage a; versus input voltage e will be modified to a form similar to that shown in Fig. 3. This curve has a straight-line portion 20 identical to that present when there is no constant frequency voltage, and a curved portion 26 rounding out the lower end of the curve. In this case the diode will start to conduct at a voltage 27 equal to the difference between the bias voltage and the peak constant frequency voltage. The modified portion of the. curve extends to the point 29 corresponding to an input voltage 28 equal to the sum of a bias voltage 21 and the peak constant frequency voltage. The shape of this lower portion of the curve may be varied as shown in Fig. 4 by changing the wave form of the constant frequency voltage. When a square wave is used as a constant frequency voltage, the lower end of the curve will be a straight line 30. When the constant frequency voltage is triangular or saw-toothed, the lower end of the curve will be a square-law function 31. When a sine-wave voltage is used as the constant frequency signal, the characteristic curve will be curved but will not have as great a curvature as when a triangular wave form is used. In order for the output signal to correspond accurately to the desired function of the input signal, it is necessary that the constant frequency voltage have a frequency'substantially greater than the maximum frequency of the input voltage. Good results have been obtained when the constant frequency was at least ten times the input frequency, although somewhat lower frequencies may be satisfactory in some instances.

" One method of building a simple curve, according to the principles of this invention, is illustrated in the circuit of Fig. 5. This circuit differs from the basic circu t of Fig. 1 only in that a diode 40 and a biasing battery 41 are connected between ground and some point along,

the series resistance 14. A function generated by such a circuit is illustrated in Fig. 6A. The dashed lines in this figure indicate the function generated without introducing the constant frequency voltage. The first derivative of this curve appears as curve 51 in Fig. 6B. It is to be noted that this is a continuous curve. The

firs-t derivative of the linear function produced without the constant frequency voltage is illustrated by the dashed lines 52 and appears as a step function. The second differential of the function 50 of Fig. 6A is illustrated in Fig. 6-C as curve 53. It is to be noted that this curve is also a continuous function. The second derivative of the straight-line function as shown by the dashed lines of Fig. 6A is indicated by the pulses54 and 55 of Fig. 6C. These pulses would be practically useless in any application where the second derivative of the generated function is desired.

Another method of building up functions with the generator of this invention is illustrated in the schematic diagram of Fig. 7. Here the series combination of the source of constant frequency voltage 11, biasing battery 12, diode 13, and series resistance 14 is placed in parallel with a second similar series circuit consisting of a second source of constant frequency potential 60, a biasing battery 61, a diode 62, and a series resistance 63. By combining the two series circuits in this manner, the resultant output from terminal 18 is the sum of the functions generated by each series circuit. A typical curve generated by such a circuit is illustrated in Fig. 8. Without a constant frequency voltage, the second series circuit generates a straight-line function 71 with a slope proportional to the ratio of the feed-back resistance 16 to the series resistance 63. The second biasing battery has a potential 70, and the portion of the curve modified by the constant frequency voltage appears between the points corresponding to input voltages equal to the sum 72 and difference 73 of the second biasing battery 71) and the peak voltage of the second source of constant frequency voltage oil.

In this manner, more complex functions may be generated to give the desired relationship between the output and the input voltage. Additional series circuits may be added in parallel to the system shown in Fig. 7; diode limiters may be added to these series circuits in the manner illustrated in Fig. 5 or various combinations of these systems may be employed, and it is obvious that other variations not illustrated herein may also be used without departing from the scope of this invention.

As an example, the circuit of Fig. 9 illustrates a method of generating a parabolic function. Here thesecond series circuit includes an amplifier 80. This amplifier has an amplification factor of 1, the output being the inverse of the input. In other words, an increase in the input voltage applied to terminal 10 and appearing at the input terminal of constant frequency source of potential 11 will appear as a decrease of the same magnitude to the input of the terminals of the source of constant frequency potential 60. The function generated by this circuit is illustrated in the curve of Fig. 10. The function of positive input voltage is generated by the first series circuit, while the function of negative input voltages 91 is generated by the second series circuit. Since the two halves of the parabola are identical, the same series resistance 14 may be used for both. In order for the curves 90 and 91 to coincide, the bias batteries 12 and 61 have voltages equal in magnitude to the peak constant frequency voltages 92 and 93, respectively, applied to the circuit, and the two constant frequency voltages are equal in magnitude.

An alternative system for the electronic generation of a parabolic function is illustrated in the schematic diagram of Fig. 11. In this circuit, an input voltage is applied to terminal and a constant frequency voltage is applied to terminal 101. The constant frequency voltage 111 has a DC. level 112, equal to the peak value of the constant frequency voltage, so that the peaks will not cause conduction through the diodes until a signal is present at terminal 100 and that one of the diodes will conduct when a signal appears at terminal 100. These two voltages are added together and their sum appears at a point 104 beyond isolating resistances 102 and 103. The input signal is also applied to an amplifier 105 having an amplification factor of -l. The output of the amplifier is similarly added to the constant frequency voltage to give a sum of these two voltages at a point 106 beyond isolating resistances 107 and 108. When the voltage input to terminal 100 is positive, a diode 109 whose plate is connected to a point 104 will conduct, thereby producing an output signal at terminal 18, which is a parabolic function of the input signal to terminal 100. This function will be similar to the curve corresponding to positive values of input voltage in the curve of Fig. 10. When the input signal to terminal 100 is negative, the diode 110 Whose plate is connected to point 106 and whose cathode is connected to series resistance 14 will conduct, resulting in an output signal at terminal 18 similar to the functions of negative input voltage appearing in Fig. 10. 1

An electronic multiplier utilizing the principle of this invention is illustrated in the schematic diagram of Fig. 12. Basically, this circuit consists of the parabolic function generator illustrated in Fig. 11 for generating positive output voltages connected to a similar circuit for generating negative output voltages. Combining two parabolic generators as the inputs of a single function generator in a manner which yields an output proportional to the difference between the functions generated by the two inputs provides a means of producing a fourquadrant multiplier by the well-known quarter-squaredifference method. In this circuit, two variable voltages X and Y which are to be multiplied together are connected respectively to terminals 120 and 121. A voltage which corresponds to the inverse of X is provided at the output of amplifier 122 whose input is connected to terminal 120, that is, an increase in the voltage X provides a decrease in voltage of the same magnitude at the output of the amplifier 122. Similarly, an amplifier 123 whose input is connected to terminal 121 provides an output voltage that is the inverse to the Y input. A constant frequency or sweep voltage similar to that described in conneciton with the circuit of Fig. 11 is applied to a terminal 124. An inverse sweep voltage is provided by means of an amplifier 125 whose input is connected to the terminal 124. These input voltages are combined re spectively in four resistance networks 126, 127, 128, and 129, Which are connected respectively to diodes 130, 131, 132, and 133. The circuit is arranged in such a manner that the voltage applied to diode 130 is equal to the sum of X and Y. The voltage applied to diode 131 is equal to the sum of the inverse of X and Y. The voltage applied to diode 132 is equal to the sum of Y and the inverse of X, and the voltage applied to diode 133 is equal to the sum of X and the inverse of Y. A constant frequency voltage is impressed on each of these sums. The cathodes of diodes 130 and 131 are connected to a series resistance 134, which is connected to the input of a feed-back amplifier 135. Similarly, the plates of diodes 132 and 133 are connected to a series resistance 136 to the input of amplifier 135.

When the sum X plus Y is positive, diode 130 will conduct and the output appearing at terminal 137 will be proportional to the quantity (X plus Y) if a triangular constant frequency voltage is applied to terminal 124. Similarly, if the sum X minus Y is negative, a voltage proportional to (X -Y) will also appear at the output terminal 137. By the formula:

these conditions will be proportional to the product of X and Y when Y is greater than X. When Y is less than X, diode 132 conducts to give a similar result. By the use of four diodes, this circuit acts as a four-quadrant multiplier, the output voltage from terminal 137 being proportional always to the product of the input voltages X and Y.

This electronic multiplier, which utilizes the squarelaw feature of the triangular constant frequency voltage, requires a minimum of equipment to function as an accurate high-speed four-quadrant multiplier.

Thus, it is seen that functions may be produced with continuous derivatives which are suitable for many applications. Foreknowledge of the shape of the output permits functions to be approximated with an accuracy greater than that of conventional diode circuits. Choice of wave form utilized permits special curves and outputs to be produced when desired with a minimum. of equipment.

It will be understood, of course, that while the forms of the invention herein shown and described constitute the preferred embodiment of the invention, it is not intended herein to illustrate all of the possible equivalent forms or ramifications of the invention. It will also be understood that the words used are words of description rather than of limitation, and that various changes, such as changes in arrangements of parts, may be substituted without departing from the spirit or scope of the invention herein disclosed.

What is claimed is:

1. A function generator comprising input terminal means for receiving an input voltage, output terminal means, an electrical network connecting said input terminal means with said output terminal means to provide an output voltage that is a smooth nonlinear function of said input voltage, said network including a generator for producing a constant frequency voltage having a frequency of about ten times the frequency of said input voltage, said generator connected to said input terminal means to produce a combined voltage of said constant frequency voltage and said input voltage, a diode connected in series with said input terminal means, biasing means providing a constant voltage bias to said diode, said constant voltage bias being greater than the peaks of said constant frequency voltage, a feedback amplifier, said diode connected in series with and between said feedback amplifier and said constant frequency generator, and feedback amplifier having a feedback network, said feedback network including a feedback resistor and a feedback capacitor connected in parallel therewith, said feedback capacitor having an electrical capacitance of such a value so as to smooth the high frequency peaks of said combined voltage, and the output of said feedback amplifier being connected to said output terminal means so as to apply thereto an output voltage which is a nonlinear function of said input voltage.

2. A function generator as set forth in claim 1 having a resistance connecting said diode with said feedback amplifier, and another diode connected across the input of said feedback amplifier, with one end thereof connected intermediate said resistance and another biasing means connected in parallel with said feedback amplifier to bias said other diode.

References Cited in the file of this patent UNITED STATES PATENTS Lakatos Apr. 6, 1954 OTHER REFERENCES it is seen that the output appearing at terminal 137 under tronics, November 1952, pages 122-126. 

